Dividing integers

The sign rules for division are the same as for multiplication. The only new thing to remember: never divide by zero.

Idea

Sign rules.

• Positive $\div$ Positive = Positive. $20 \div 5 = 4$.
• Positive $\div$ Negative = Negative. $20 \div (-5) = -4$.
• Negative $\div$ Positive = Negative. $(-20) \div 5 = -4$.
• Negative $\div$ Negative = Positive. $(-20) \div (-5) = 4$.

Same signs $\to$ positive; different signs $\to$ negative.

Why? Division undoes multiplication. If $(-5) \times 4 = -20$, then $(-20) \div 4 = -5$ and $(-20) \div (-5) = 4$. The sign rules of multiplication automatically give the rules of division.

Special cases.

• $0 \div a = 0$ for any $a \neq 0$ (zero divided by anything is zero).
• $a \div 0$ is undefined. You cannot divide by zero , not even $0 \div 0$.
• $a \div 1 = a$ (dividing by $1$ changes nothing).
• $a \div (-1) = -a$.

Closure. Integer division is not always closed. $7 \div 3$ is not an integer. But when the division is exact (no remainder), the result is an integer.

Order matters. $a \div b \neq b \div a$ in general. So division is not commutative or associative.

Worked examples

Example 1. Compute $(-72) \div 9$.

Different signs $\Rightarrow$ negative. $72 \div 9 = 8$. Result: $-8$.

Example 2. Compute $(-56) \div (-7)$.

Same signs $\Rightarrow$ positive. $56 \div 7 = 8$. Result: $+8$.

Example 3. Compute $40 \div (-8)$.

Different signs $\Rightarrow$ negative. $40 \div 8 = 5$. Result: $-5$.

Example 4. What is $0 \div (-15)$?

$0$ divided by anything non-zero is $0$. Result: $0$.

Try it yourself

1. Compute $(-48) \div 6$.
2. Compute $(-100) \div (-25)$.
3. Compute $36 \div (-4)$.
4. Compute $0 \div (-9)$.
5. Compute $(-1) \div (-1)$.
6. Is $(-5) \div 0$ defined?
7. Compute $(-144) \div (-12)$.
8. The temperature dropped by $30^\circ$ in $6$ hours. Average drop per hour?

Activity

Make a sign-rule table for division on a small flashcard. Glue it on the inside cover of your notebook for quick reference.